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Proyecciones (Antofagasta)

Print version ISSN 0716-0917

Proyecciones (Antofagasta) vol.36 no.2 Antofagasta June 2017 


Fuzzy normed linear space valued sequence space

Paritosh Chandra Das1 

1 Rangia College, India , e-mail :


In this article we define the notion of fuzzy normed linear space valued sequence space in a fuzzy normed linear space X and discuss some of its properties like completeness, monotone, solidness, convergence free and symmetricity. Also we prove some inclusion results.

Key Words: Fuzzy real number; fuzzy normed linear space; monotone; solid space; convergence free and symmetricity

1. Introduction

The concept of fuzzy set, a set whose boundary is not sharp or precise has been introduced by L. A. Zadeh in 1965. It is the origin of new theory of uncertainty, distinct from the notion of probability. After the introduction of fuzzy sets, the scope for studies in different branches of pure and applied mathematics increased widely. The notion of fuzzy set theory has been applied to introduce the notion of fuzzy real numbers which helps in constructing the sequence of fuzzy real numbers. Different types of sequence spaces of fuzzy real numbers have been studied under classical metric by Das (1), (2), Matloka (5),Subrahmanyam (6), Tripathy et al (7), Tripathy and Dutta (8), (9), Tripathy and Sarma (10), Tripathy and Debnath (11) and many others. A few works on fuzzy norm which has relationship with fuzzy metric have been done by Felbin (3) and some others. There is a lot of material for the classes of sequences those can be examined by fuzzy norm.

2. Definitions and preliminaries

A fuzzy real number X is a fuzzy set on R, i.e. a mapping X: R →I (=0,1) associating each real number t with its grade of membership X (t).

A fuzzy real number X is called convex if min

If there exists then the fuzzy real number X is called normal.

A fuzzy real number X /is said to be upper-semi continuous if, for each is open in the usual topology of R.

The set of all upper-semi continuous, normal, convex fuzzy real numbers is denoted by R (I). Throughout the article, by a fuzzy real number we mean that the number belongs to R (I).

The α -level set defined as then it is the closure of the strong O-cut. (The strong α- cut of the fuzzy real number X, for is the set

Let Then the arithmetic operations on R (I) in terms of α-level sets are defined as follows:

The absolute value, is defined by (see for instance Kaleva and Seikkala ((4)))

A fuzzy real number . The set of all non-negative fuzzy real numbers is denoted by R* (I).

Fuzzy Normed Linear Space

Let X be a linear space ove R and the mapping and the mappings, be symmetric, non-decreasing in both arguments and satisfy Write and suppose for all

The quadruple is called a fuzzy normed linear space and a fuzzy norm on the linear space X, if

In the sequel we take for or simply by 𝑋 in this case.

With these 0, 1, we have (refer to Felbin 3) in a fuzzy normed linear space the triangle inequality (iii) of the definition of fuzzy normed linear space is equivalent to

The set of all sequences in a vector space X is a vector space with respect to pointwise addition and scalar multiplication. Any subspace is a fuzzy normed linear space, then is called a fuzzy normed linear space-valued sequence space.

Throughout fnls denotes fuzzy normed linear space.

A fnls-valued sequence space E F (X) is said to be normal (or solid) if be a fnls-valued sequence space. A K- step space of

A canonical pre-image of a sequence is a sequence defined as follows:

A canonical pre-image of a step space is a space of canonical pre-images of all elements in is in canonical pre-image if and only if 𝑦 is canonical pre-image of some .

A fnls-valued sequence space E F (X) is said to be monotone if E F (X) contains the canonical pre-images of all its step spaces.

From the above definitions we have following remark.

Remark 2.1. A fnls-valued sequence space is monotone.

A fnls-valued sequence space E F (X) is said to be symmetric is a permutation of N.

A fnls-valued sequence space E F (X) is said to be convergence free if whenever

A fuzzy normed linear space is called fuzzy complete if every Cauchy sequence in X converges in X

With the concept of fuzzy norm the fuzzy normed linear space valued sequence space is defined by:

Clearly, is a norm.

Throughout denote the spaces of all, p-absolutely summable and convergent sequences in fuzzy normed linear space X respectively.

3. Main results

Theorem 3.1. The class of p-absolutely summable sequences is fnls-valued sequence space.

Theorem 3.2. The space is complete with the norm

Theorem 3.3. The sequence space is solid and as such is monotone.

Proof: Consider two sequences

Theorem 3.4. The sequence space is not convergence free.

Proof: The result follows from the following example.

Example 3.1. Consider the sequence defined as follows.

We have X is a fuzzy normed linear space. For any sequence let us consider defined as follows:

Then for k even and using (3.1), we have

Theorem 3.5. The sequence space is symmetric.


[1] Das, P. C.: Statistically convergent fuzzy sequence spaces by fuzzy metric; Kyungpook Math. Journal, 54 (3), pp. 413-423, (2014). [ Links ]

[2] Das, P. C.: p-absolutely Summable Type Fuzzy Sequence Spaces by Fuzzy Metric; Boletim da Sociedade Paranaense de Matemtica, 32 (2), pp. 35-43, (2014). [ Links ]

[3] Felbin, C.: Finite dimensional fuzzy normed linear space; Fuzzy Sets and Systems, 48, pp. 239-248, (1992). [ Links ]

[4] Kelava, O. and Seikkala, S.: On fuzzy metric spaces; Fuzzy Sets and Systems, 12, pp. 215-229, (1984). [ Links ]

[5] Matloka, M.: Sequences of fuzzy numbers ; BUSEFAL, 28, pp. 28-37, (1986). [ Links ]

[6] Subrahmanyam, P. V. : Cesaro Summability for fuzzy real numbers; J. Analysis, 7, pp. 159-168, (1999). [ Links ]

[7] Tripathy, B. C. Baruah, A. Et, M. and Gungor, M. : On almost statistical convergence of new type of generalized difference sequence of fuzzy numbers; Iranian Journal of Science and Technology, Transacations A Science, 36 (2), pp. 147-155, (2012). [ Links ]

[8] Tripathy, B. C. and Dutta, A. J. : Statistically pre-Cauchy fuzzy real-valued sequences defined by Orlicz function; Proyecciones Journal of Mathematics, 33 (3), pp. 235-243, (2014). [ Links ]

[9] Tripathy, B. C. and Dutta, A. J. : Lacunary I-convergent sequences of fuzzy real numbers; Proyecciones Journal of Mathematics, 34 (3), pp. 205-218, (2015). [ Links ]

[10] Tripathy, B. C. and Sarma, B. : Sequence spaces of fuzzy real numbers defined by Orlicz functions; Math. Slovaca, 58 (5), pp. 621-628, (2008). [ Links ]

[11] Tripathy, B. C. and Debnath, S. : On generalized difference sequence spaces of fuzzy numbers; Acta Scientiarum Technology, 35 (1), pp. 117-121, (2013). [ Links ]

Received: February 2016; Accepted: March 2017

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